All Questions
22
questions
5
votes
1
answer
234
views
How does one rigorously define two-point functions?
Let $\mathscr{H}$ be a complex Hilbert space, and $\mathcal{F}^{\pm}(\mathscr{H})$ be its associated bosonic (+) and fermionic (-) Fock spaces. Given $f \in \mathscr{H}$, we can define rigorously the ...
- 1,131
17
votes
2
answers
1k
views
What physical processes other than scattering are accounted for by QFT? How do they fit into the general formalism?
For background, I'm primarily a mathematics student, studying geometric Langlands and related areas. I've recently been trying to catch up on the vast amount of physics knowledge I'm lacking, but I've ...
- 171
1
vote
0
answers
40
views
Prefactor to amplitude for massless fermions/massive bosons
I was reviewing some of my old notes and found this formula for a matrix elements between two states:
$$
<f|S|i>= \delta_{fi} + [(2\pi)^4 \delta^4(P_f - P_i) \prod_{ext. fermion}\left(\frac{m}{...
1
vote
1
answer
120
views
Understanding from $S$-Matrix to Feynman-Rules in scalar QFT [closed]
I am learning QFT at the moment and the process from defining the S-Matrix to deriving the feynman rules is in my opinion pretty complicated, since there are many different things to pay attention to. ...
- 505
4
votes
2
answers
161
views
Are amplitudes for inverse processes related to each other?
The (generalized) optical theorem is presented in the book of Peskin and Schroeder (An introduction to Quantum Field Theory - chapter 7-Radiative Corrections:Some Formal Developments) as follows
$-i (\...
1
vote
1
answer
137
views
Question about a completeness relation in Peskin Chapter 7: radiative corrections p. 212 equation 7.2
In P&S book, in Charpter 7: radiative corrections p. 212 （attached beneath, the completeness relation: (Equation 7.2)
$$\textbf{1}=\left\vert \Omega\right\rangle\left\langle\Omega\right\vert+\sum\...
- 73
4
votes
1
answer
172
views
How to obtain (interacting) time-ordered correlation functions from the S-matrix - reverse of the LSZ formula?
The LSZ formula shows how to obtain the S-matrix elements from the time-ordered correlation functions of the interacting fields.
I wonder if there is a reverse formula; that is, can we find the ...
- 1,669
0
votes
2
answers
330
views
Peskin & Schroeder LSZ formula missing in- and out states
In Peskin and Schroeder the LSZ-formula is given as below where the states in the $S$-matrix element are fully interacting Heisenberg states.
$$\begin{array}{l}\prod_{1}^{n} \int d^{4} x_{i} e^{i p_{i}...
- 498
3
votes
1
answer
255
views
Field strength renormalization constant Peskin and Schroeder
In chapter 7 of Peskin and Schroeder, on page 214 we have the expression of the spectral density function
$$\tag{7.7}\rho(M^2)=\sum_{\lambda} (2\pi)\delta(M^2-m_\lambda^2)|\langle\Omega|\phi(0)|\...
- 789
4
votes
1
answer
275
views
LSZ reduction formula vs Dyson's expansion
In quantum field theory, we have use perturbation series to compute the $S$-matrix elements. For example:
$$S=1+\sum_{i=1}^\infty\frac{(-i/\hbar)^n}{n!}\int_{-\infty}^\infty...\int_{-\infty}^\infty T[...
- 789
1
vote
1
answer
226
views
Different forms of the LSZ reduction formula
I'm studying Chapter 7 section 2 of Peskin and Schroeder on the LSZ reduction formula, on page 227 they write the LSZ reduction formula
$$\tag{7.42}\prod_1^n \int d^4x_i e^{ip_i\cdot x_i}\prod_1^m\int ...
- 789
4
votes
2
answers
281
views
Derivation in LSZ Reduction Formula
In deriving the LSZ formula, a crucial step is to show
$$\langle|a_{p}^{\dagger}|\rangle=-i\int dx^0 \int \mathrm{d}^{3} x \partial_{0}\langle | e^{-i p\cdot x} \overleftrightarrow{\partial_{0}} \phi(...
- 381
4
votes
1
answer
419
views
Resolution of the Identity in Quantum Field Theory
In Peskin and Schroder's QFT book, on page 212, eq.7.2, they use the completeness relation in a derivation involving the two-point correlation function:
$$
\mathbf{1}=|\Omega\rangle\langle\Omega|+\...
- 1,421
2
votes
1
answer
384
views
The relation between full Green's function and S-matrix
I'm learning Green's function in condensed matter. The full Green's function is defined as
$$G(k_2,t_2;k_1,t_1) = \langle\Omega |T a_{k_1}(t_1)a_{k_2}^{\dagger}(t_2) |\Omega \rangle $$
The $\Omega$ is ...
- 169
3
votes
1
answer
1k
views
LSZ formula for initial and final one particle states
The LSZ formula for a real scalar field $\varphi$ is (Srednicki 5.24)
$$
\left<f|i\right>=i^{n+n'}\int d^4x_1e^{ik_1x_1}(-\partial_1^2+m^2)...\\
\quad d^4x'_1e^{ik'_1x'_1}(-\partial_{1'}^2+m^2).....
- 93